Questions
1 question per paper
Difficulty
Medium
Importance
Medium yield for HPCL/NTPC/BHEL
Overview
Complex variables involve functions of the form f(z) = u + iv, which extend standard real calculus into the complex plane. This topic is essential for PSU exams as it provides the foundation for solving contour integrals and series expansions that frequently appear in engineering mathematics sections. Aspirants should focus on identifying analyticity and applying integral theorems efficiently.
Analytic Functions & Cauchy-Riemann
A function is analytic at a point if it is differentiable in a neighborhood of that point. The Cauchy-Riemann equations provide the necessary and sufficient conditions for a function to be analytic in a given domain.
- f(z) = u(x,y) + i*v(x,y)
- Cauchy-Riemann equations: du/dx = dv/dy and du/dy = -dv/dx
- Harmonic condition: d^2u/dx^2 + d^2u/dy^2 = 0
- Analytic functions satisfy Laplace's equation
- Necessary condition: Partial derivatives must be continuous
Cauchy's Integral Theorem & Formula
These theorems relate the line integral of an analytic function around a closed contour to the values of the function inside or on the contour. Most exam questions require evaluating integrals by identifying singular points within the contour.
- Cauchy's Integral Theorem: Integral f(z) dz = 0 for analytic functions over closed path
- Cauchy's Integral Formula: f(a) = (1/2*pi*i) * Integral [f(z)/(z-a)] dz
- General Formula: f^n(a) = (n!/2*pi*i) * Integral [f(z)/(z-a)^(n+1)] dz
- Closed path integration yields 0 if no poles are enclosed
- Residue theorem usage for poles inside contours
Taylor & Laurent Series
Power series expansions are used to represent complex functions near points where they are analytic or near singularities. Taylor series is for analytic regions, while Laurent series accounts for regions containing poles.
- Taylor series: f(z) = sum of a_n(z-z0)^n
- Laurent series: f(z) = sum of a_n(z-z0)^n + sum of b_n(z-z0)^(-n)
- Principal part: The sum of terms with negative powers
- Singularity types: Removable, Pole, and Essential
- Radius of convergence is distance to nearest singularity
Formula Sheet
du/dx = dv/dy
du/dy = -dv/dx
f(a) = (1/2*pi*i) * Integral [f(z)/(z-a)] dz
f^n(a) = (n!/2*pi*i) * Integral [f(z)/(z-a)^(n+1)] dz
Residue at pole z=a of order m: (1/(m-1)!) * lim(z->a) [d^(m-1)/dz^(m-1) (z-a)^m * f(z)]
Exam Tip
Always locate the singularities (poles) first and check if they lie inside the given contour before attempting complex integration.
Common Mistakes
- Failing to verify if the function is analytic inside the contour before applying Cauchy's Integral Theorem.
- Incorrectly identifying the order of the pole when using the Residue Theorem.
- Confusing the signs in the Cauchy-Riemann equations during derivation.
More Revision Notes
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